How Function Composition Works
Composition chains functions like a production line: g processes x first, then f processes g’s output. The notation (f∘g)(x) is read “f of g of x”, and the rightmost function always acts first.
Substitution, Not Multiplication
The classic mistake is multiplying the functions instead of substituting. For f(x) = 2x + 1 and g(x) = x², (f∘g)(x) replaces every x in f with x², giving 2x² + 1 — not 2x³ + x².
Order Matters
(g∘f)(x) for the same pair is (2x+1)² = 4x² + 4x + 1 — completely different. Always substitute in the order the notation demands, innermost first.
Composition is how inverses are verified: try our inverse function calculator or simplify the substituted expressions with the algebra calculator.